Problem 39 - Calculus of a Single Variable

Question

Horizontal Asymptotes In Exercises \(39-42,\) use a graphing utility to graph the function and identify any horizontal asymptotes. $$ f(x)=\frac{|x|}{x+1} $$

Step-by-Step Solution

Answer

The horizontal asymptote of the function \(f(x)=\frac{|x|}{x+1}\) is \(y=1\).

Step 1: Understanding the function

A starting point would be to study the nature and properties of the function \(f(x)=\frac{|x|}{x+1}\). Notice that it is a rational function where the numerator is the absolute value of \(x\) and the denominator is \(x+1\).

Step 2: Determine horizontal asymptotes

Horizontal asymptotes of a rational function are determined using the degrees of the numerator and denominator. If the degree of the numerator and denominator is the same, the horizontal asymptote will be the quotient of the leading coefficients. In this case, the leading coefficients of both the numerator and denominator are 1 and hence the horizontal asymptote is the line \(y=1\).

Step 3: Confirm by graphing

Now graph the function and confirm whether our identified horizontal asymptote (\(y=1\)) exists. You should notice that the graph approaches the line \(y=1\) as \(x -> ∞\) and \(x -> - ∞\), confirming that it is indeed the horizontal asymptote.

Key Concepts

Rational Functions

Rational functions are mathematical expressions that take the form of a fraction. In these fractions, both the numerator and the denominator are polynomials. This means the top and bottom parts of the fraction involve variables raised to whole number powers—and they can be simple or complex depending entirely on the equation in question. Take the function from the exercise, for example: \( f(x) = \frac{|x|}{x+1} \). Here, the numerator is the absolute value of \( x \), and the denominator is \( x+1 \). In rational functions, one important aspect we often look for is horizontal asymptotes. These are horizontal lines that the graph of the function approaches as \( x \) moves towards positive or negative infinity.

• If the degrees of the numerator and the denominator are the same, the horizontal asymptote is found at the ratio of the leading coefficients.
• If the degree of the numerator is smaller, the asymptote is always at \( y=0 \).

Understanding these rules helps in analyzing the behavior of rational functions and predicting their graphs.

Absolute Value

The absolute value term |x| means a key concept here; it refers to the distance a number is from zero on the number line and is always non-negative. Simply put, \( |x| = x \) if \( x \geq 0 \) and \( |x| = -x \) if \( x < 0 \).For the function \( f(x) = \frac{|x|}{x+1} \), this characteristic of absolute value influences the overall behavior of the function. It means that for positive values of \( x \), the function behaves regularly, keeping the numerator as it is. However, for negative \( x \), the absolute value makes the numerator positive, changing how the function approaches its horizontal asymptote.Understanding and working with absolute values are essential, especially when dealing with rational expressions, as it can affect the calculation of limits and asymptotes significantly.

Graphing Utility

Graphing utilities, like graphing calculators or software tools, are incredibly helpful when analyzing complex functions. They provide visual insights that assist in identifying characteristics like horizontal asymptotes, turning points, or intercepts easily.When we talk about the function \( f(x) = \frac{|x|}{x+1} \), using a graphing utility helps confirm the theoretical analysis. It visualizes how the graph approaches the horizontal asymptote \( y=1 \), and it graphically represents the effect of the absolute value on the rational function.Using these tools, students learn skills like:

• Plotting rational functions accurately and efficiently.
• Understanding the accuracy of theoretical findings through visual confirmations.
• Experimenting with different values or similar functions to discover patterns or irregularities.

With graphing utilities, students can enhance their understanding of both basic and advanced function evaluations, making complex equations significantly more manageable.